Finite Group Databasebeta

← #34099398

Subgroups — #34099398

Order 20096.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×D628C211yesmaximal, characteristic
2#23757669C211yesmaximal, characteristic, complemented
3#23757475C211yesmaximal, characteristic
4#23757020C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic
7#475×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157—21no—
9C2×#14578941—21no—
10#19097771—21no—
11#19097771C2211yescharacteristic
12#19097751—22no—
13#19097751—21no—
14C4×D628—21no—
15C4×D628C2211yescharacteristic
16#19097702—22no—
17C4⋊#11929434C2211yescharacteristic
18#19097695—21no—
19C4×#11929434C2211yescharacteristic
20C4⋊Q8×C157C2211yescharacteristic
21C4×Q8×C157C2211yescharacteristic
22#106×C157—21no—
23C4⋊C8×C157C2211yescharacteristic
24#103×C157—21no—
25C2×Dic314—22no—
26C2×Dic314—21no—
27Q8×D157—42no—
28Q8×D157—42no—
29C2×D628D411yescharacteristic
30C2×C4×D157—22no—
31C2×C4×D157—21no—
32#14578941—42no—
33C2×#11929434—21no—
34C2×#11929434—21no—
35C4⋊Dic157—22no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38C4⋊Dic157D412yes—
39#14578931—22no—
40#14578931—21no—
41C2×Q8×C157—21no—
42C2×Q8×C157D411yescharacteristic
43C2×C8×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157—21no—
46C4⋊C4×C157—21no—
47C4⋊C4×C157D411yescharacteristic
48C42×C157—21no—
49C42×C157C2311yescharacteristic
50C22×D157—21no—
51Dic314—44no—
52Dic314—41no—
53Dic314—41no—
54D628—21no—
55D628SD1612yes—
56C2×Dic157—22no—
57C2×Dic157—21no—
58C4×D157—44no—
59C4×D157—41no—
60C4×D157—41no—
61#11929434—22no—
62#11929434—41no—
63Q8×C157—21no—
64Q8×C157—41no—
65Q8×C157—41no—
66Q8×C157SD1612yescomplemented
67C2×C4×C157—21no—
68C2×C4×C157—21no—
69C2×C4×C157—41no—
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157C2×D411yescharacteristic
72C2×C4×C157C2×D411yescharacteristic
73C8×C157—41no—
74C22×C157#12011yescharacteristic
75C2×D157—22no—
76C2×D157—41no—
77C2×D157—41no—
78C4×C157—22no—
79C4×C157—21no—
80C4×C157—41no—
81C4×C157—81no—
82C4×C157C2×SD1611yescharacteristic
83C4×C157C2×SD1611yescharacteristic
84C4×C157AGL(1,ℤ/8)11yescharacteristic
85C4×C157#13711yescharacteristic
86Dic157—42no—
87Dic157—41no—
88C2×C157#44811yescharacteristic
89C2×C157#45011yescharacteristic
90C2×C157#45611yescharacteristic
91D157—42no—
92C157#165111yescharacteristic, complemented, Sylow
93#1651—1571nomaximal, Sylow
94D4×Q8—1571no—
95#498—1571no—
96#475D15711yescharacteristic, complemented
97#461—1571no—
98#332—1571no—
99Q8⋊C8—1571no—
100D4⋊C8—1571no—
101C22×Q8—3141no—
102C2×SD16—3141no—
103C4⋊Q8—3141no—
104C4⋊Q8—1571no—
105C4⋊Q8C2×D15711yescharacteristic
106C22⋊Q8—3142no—
107C22⋊Q8—3141no—
108C4×Q8C2×D15711yescharacteristic
109C4×D4—3141no—
110C4×D4—1571no—
111#106—21no—
112#106—3142no—
113C4⋊C8—1571no—
114C4⋊C8C2×D15711yescharacteristic
115#103—21no—
116#103—3141no—
117C4×C8—1571no—
118C2×Q8—21no—
119C2×Q8—3142no—
120C2×Q8—3141no—
121C2×Q8—6282no—
122C2×Q8—6282no—
123C2×Q8C22⋊D15711yescharacteristic
124C2×D4—1571no—
125C22×C4—3142no—
126C22×C4—3141no—
127SD16—6282no—
128C2×C8—21no—
129C2×C8—3141no—
130C2×C8—3141no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3142no—
135C4⋊C4—3141no—
136C4⋊C4—1571no—
137C4⋊C4—1572no—
138C4⋊C4C22⋊D15711yescharacteristic
139C22⋊C4—3142no—
140C22⋊C4—3141no—
141C42—21no—
142C42C22×D15711yescharacteristic
143C23—3141no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—6284no—
148Q8—6281no—
149Q8—6281no—
150Q8#1457894112yescomplemented
151D4—3141no—
152D4—1572no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—3142no—
157C2×C4—3141no—
158C2×C4—6284no—
159C2×C4—6281no—
160C2×C4—6281no—
161C2×C4D4×D15711yescharacteristic
162C2×C4D4×D15711yescharacteristic
163C2×C4C2×C22⋊D15711yescharacteristic
164C8—41no—
165C8—3142no—
166C8—6281no—
167C22—3142no—
168C22—6281no—
169C22—6281no—
170C22#1909773911yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—6282no—
176C4—6281no—
177C4SD16×D15711yescharacteristic
178C4C2×#1457894111yescharacteristic
179C4#1909779611yescharacteristic
180C4#1909780011yescharacteristic
181C2—6282no—
182C2#2375741011yescharacteristic
183C2#2375742311yescharacteristic
184C2#2375745411yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#475×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#475D157
hypocenterC157#1651
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #23757669C2semidirect
4. #23757020C2semidirect
5. Q8⋊#11929434C2semidirect
7. #475×C157C2semidirect
66. Q8×C157SD16semidirect
92. C157#1651semidirect
96. #475D157semidirect
150. Q8#14578941semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
Q8×D6282
#237576692
#237574752
#237570202
Q8⋊#119294342
#237570002
#475×C1572
#1651157